Basics of sequences and series and it is very use full to engineering student
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Se uences and Series D Sequences and Series By Venkateaswara rao.G
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Se uences and Series Examples of Sequences A sequence is an ordered list of numbers The 3 dots are used to show that a sequence continues
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Se uences and Series Recurrence Relations Can you predict the next term of the sequence Suppose the formula continues by adding 2 to each term. The formula that generates the sequence is then where un and u are terms of the sequence ul is the 1st term, SO ul = 3 n = I = UI +2 u n = 2 =U2+2 u etc.
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Se uences and Series Recurrence Relations A formula such as un+l = +2 is called a recurrence relation e.g. 1 Give the 1st term and write down a recurrence relation for the sequence 4, 16, — 64, Solution : 1st term: Recurremce relation: un+l Other letters may be used instead of u and n, so the formula could, for example, be given as
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Se uences and Series Recurrence Relations e.g. 2 Write down the 2nd 3rd and 4th terms of the sequence given by ul = 5, u Solution: The se uence is 7, 11, 19,
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Se uences and Series Properties of sequences Convergent sequences approach a certain value approaches 2 16
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Se uences and Series Properties of sequences Convergent sequences approach a certain value approaches O This convergent sequence also oscillates
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Se uences and Series Properties of sequences Divergent sequences do not converge 2, 4, 6, 8, 10,
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-c Se uences and Series Properties of sequences Divergent sequences do not converge This divergent sequence also oscillates
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Se uences and Series Properties of sequences Divergent sequences do not converge u n This divergent sequence is also periodic
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Se uences and Series Convergent Values It is not always easy to see what value a sequence converges to. e.g. u 11 The sequence is 1, 7, 7 10-3un u 103 11 To find the value that the sequence converges to we use the fact that eventually ( at infinity! ) the ( n + 1 ) th term equals the n th term. 10-3u Then. u = = un = u u : u 2 = 10—3u u +3u—10=0 Multiply by u since 2
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Se uences and Series Exercises 1. Write out the first 5 terms of the following sequences and describe the sequence using the words convergent, divergent, oscillating, periodic as appropriate Ans: (b) Ans: -2, (C) ul = 16 Ans: 16 and and 8 Divergent 1 u u 2, —4, —2 Divergent Periodic 1 u —2, 1 Convergent Oscillating 2. What value does the sequence given by ul = 2, un+l +3 converge to? Let u u = 0-3u+3 u = 7
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Se uences and Series General Term of a Sequence Some sequences can also be defined by giving a general term. This general term is usually called the nth term. 1, —4, 16, The general term can easily be checked by substituting n = 1, n = 2, etc.
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1. 2. Se uences and Series Exercises 81, — 243, Write out the first 5 terms of the following sequences 2, 4, -8, 16, 2, 8, 18, 32, 50 Give the general term of each of the following sequences un =2n-1 (b) I, 4, 9, 16, 3 9, -27, 25,
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Series When the terms of a sequence are added, we get a series The sequence 1, 4, 9, 16, 25, gives the series 1 + 4 + 9 + 16 + 25 + Sigma Notation for a Series A series can be described using the general term 1 4 9 + 16 + 25 + + 100 can be written 1 is the Greek capital letter S, used for Sum
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Se uences and Series Exercises Write out the first 3 terms and the last term of 1. the series given below in sigma notation 20 . + 39 1 100 (b) 2(-3) 1 n = 20 100 2. Write the following using sigma notation (b) 2+4+8+ + 16 = 1 10 + 1024 = 1
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Se uences and Series Thankyou by Venkat
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